We propose a Bayesian approach where the signal structure can be represented by a mixture model with a submodular prior. We consider an observation model that leads to Lipschitz functions. Due to its combinatorial nature, computing the maximum a posteriori estimate for this model is NP-Hard, nonetheless our converging majorization-minimization scheme yields approximate estimates that, in practice, outperform state-of-the-art modular prior. We consider an observation model that leads to Lipschitz functions. Due to its combinatorial nature, computing the maximum a posteriori estimate for this model is NP-Hard, nonetheless our converging majorization-minimization scheme yields approximate estimates that, in practice, outperform state-of-the-art methods

why submodularity is a big deal, from the paper:

Submodularity
is considered the discrete equivalent of convexity in the sense that
submodular function minimization (SFM) admits efficient algorithms,
with best known complexity of O(N5T+N6), where T is the function
evaluation complexity [11]. In practice, however, the minimum-norm point
algorithm is usually used whenever, the minimum-norm point algorithm is
usually used, which commonly runs in O(N2), but has no known complexity
[12]. Furthermore, for certain functions which are “graph representable”
[13, 14], SFM is equivalent to the minimum s-t cut on an appropriate
graph G(V,E), with time complexity 1 O(|E|min{|V|2/3,|E|1/2}) [15].